# Enforcing geometric constraints of virtual normal for depth prediction

arXiv: https://arxiv.org/abs/1907.12209 · HTML: https://mlsanity.com/paper/1907.12209/

## Key facts

- Cited by 234 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jul 2019.

## Built on by (most cited first, 30 of 234)

- [Vision Transformers for Dense Prediction](https://mlsanity.com/paper/2103.13413/) (2021)
- [Depth Anything V2](https://mlsanity.com/paper/2406.09414/) (2024)
- [Depth Anything: Unleashing the Power of Large-Scale Unlabeled Data](https://mlsanity.com/paper/2401.10891/) (2024)
- [ZoeDepth: Zero-shot Transfer by Combining Relative and Metric Depth](https://mlsanity.com/paper/2302.12288/) (2023)
- [Metric3Dv2: A Versatile Monocular Geometric Foundation Model for Zero-shot Metric Depth and Surface Normal Estimation](https://mlsanity.com/paper/2404.15506/) (2024)
- [Metric3D: Towards Zero-shot Metric 3D Prediction from A Single Image](https://mlsanity.com/paper/2307.10984/) (2023)
- [GeoWizard: Unleashing the Diffusion Priors for 3D Geometry Estimation from a Single Image](https://mlsanity.com/paper/2403.12013/) (2024)
- [MoGe-2: Accurate Monocular Geometry with Metric Scale and Sharp Details](https://mlsanity.com/paper/2507.02546/) (2025)
- [BinsFormer: Revisiting Adaptive Bins for Monocular Depth Estimation](https://mlsanity.com/paper/2204.00987/) (2022)
- [NeW CRFs: Neural Window Fully-connected CRFs for Monocular Depth Estimation](https://mlsanity.com/paper/2203.01502/) (2022)
- [DDP: Diffusion Model for Dense Visual Prediction](https://mlsanity.com/paper/2303.17559/) (2023)
- [Transformer-Based Attention Networks for Continuous Pixel-Wise Prediction](https://mlsanity.com/paper/2103.12091/) (2021)
- [NeuRIS: Neural Reconstruction of Indoor Scenes Using Normal Priors](https://mlsanity.com/paper/2206.13597/) (2022)
- [Neural 3D Scene Reconstruction with the Manhattan-world Assumption](https://mlsanity.com/paper/2205.02836/) (2022)
- [DepthFormer: Exploiting Long-Range Correlation and Local Information for Accurate Monocular Depth Estimation](https://mlsanity.com/paper/2203.14211/) (2022)
- [P3Depth: Monocular Depth Estimation with a Piecewise Planarity Prior](https://mlsanity.com/paper/2204.02091/) (2022)
- [Boosting Monocular Depth Estimation Models to High-Resolution via Content-Adaptive Multi-Resolution Merging](https://mlsanity.com/paper/2105.14021/) (2021)
- [LocalBins: Improving Depth Estimation by Learning Local Distributions](https://mlsanity.com/paper/2203.15132/) (2022)
- [Attention Attention Everywhere: Monocular Depth Prediction with Skip Attention](https://mlsanity.com/paper/2210.09071/) (2022)
- [Monocular Depth Estimation using Diffusion Models](https://mlsanity.com/paper/2302.14816/) (2023)
- [SimpleRecon: 3D Reconstruction Without 3D Convolutions](https://mlsanity.com/paper/2208.14743/) (2022)
- [Deep Dense Multi-scale Network for Snow Removal Using Semantic and Geometric Priors](https://mlsanity.com/paper/2103.11298/) (2021)
- [DiffusionDepth: Diffusion Denoising Approach for Monocular Depth Estimation](https://mlsanity.com/paper/2303.05021/) (2023)
- [iDisc: Internal Discretization for Monocular Depth Estimation](https://mlsanity.com/paper/2304.06334/) (2023)
- [Unsupervised Scale-consistent Depth Learning from Video](https://mlsanity.com/paper/2105.11610/) (2021)
- [Ground-aware Monocular 3D Object Detection for Autonomous Driving](https://mlsanity.com/paper/2102.00690/) (2021)
- [Multi-Frame Self-Supervised Depth with Transformers](https://mlsanity.com/paper/2204.07616/) (2022)
- [Regularizing Nighttime Weirdness: Efficient Self-supervised Monocular Depth Estimation in the Dark](https://mlsanity.com/paper/2108.03830/) (2021)
- [Global-Local Path Networks for Monocular Depth Estimation with Vertical CutDepth](https://mlsanity.com/paper/2201.07436/) (2022)
- [AffordanceLLM: Grounding Affordance from Vision Language Models](https://mlsanity.com/paper/2401.06341/) (2024)

## Abstract (arXiv)

Monocular depth prediction plays a crucial role in understanding 3D scene geometry. Although recent methods have achieved impressive progress in evaluation metrics such as the pixel-wise relative error, most methods neglect the geometric constraints in the 3D space. In this work, we show the importance of the high-order 3D geometric constraints for depth prediction. By designing a loss term that enforces one simple type of geometric constraints, namely, virtual normal directions determined by randomly sampled three points in the reconstructed 3D space, we can considerably improve the depth prediction accuracy. Significantly, the byproduct of this predicted depth being sufficiently accurate is that we are now able to recover good 3D structures of the scene such as the point cloud and surface normal directly from the depth, eliminating the necessity of training new sub-models as was previously done. Experiments on two benchmarks: NYU Depth-V2 and KITTI demonstrate the effectiveness of our method and state-of-the-art performance.